PHYSICAL REVIEW B 102,245115(2020) Memory-critical dynamical buildup of phonon-dressed Majorana fermions Oliver Kaestle ,1,* Yue Sun, 2,3 Ying Hu,2,3 and Alexander Carmele 1 1Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China 3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China (Received 2 July 2020; revised 24 November 2020; accepted 24 November 2020; published 11 December 2020) We investigate the dynamical interplay between the topological state of matter and a non-Markovian dis- sipation, which gives rise to a crucial timescale into the system dynamics due to its quantum memory. We specifically study a one-dimensional polaronic topological superconductor with phonon-dressed p-wave pairing, when a fast temperature increase in surrounding phonons induces an open-system dynamics. We show that when the memory depth increases, the Majorana edge dynamics transits from relaxing monotonically to a plateau of substantial value into a collapse-and-buildup behavior, even when the polaron Hamiltonian is close to the topological phase boundary. Above a critical memory depth, the system can approach a new dressed state of the topological superconductor in dynamical equilibrium with phonons, with nearly full buildup of the Majorana correlation. DOI: 10.1103/PhysRevB.102.245115 I. INTRODUCTION phonon-renormalized Hamiltonian parameters (see Fig. 1). In contrast to Markovian decoherence that typically destroys Exploring topological properties out of equilibrium is cen- topological features for long times, we show that a finite tral in the effort to realize, probe, and exploit topological states quantum memory allows for substantial preservation of topo- of matter in the laboratory [1–15]. A paradigmatic scenario logical properties far from equilibrium, even when the polaron is where the topological system is coupled to a Markovian Hamiltonian is close to the topological phase boundary [see bath, inducing open-dissipative dynamics that is described by Fig. 2(b)]. Depending on the memory depth (i.e., the charac- aLindblad-formmasterequationforthetime-evolvedreduced teristic timescale of the quantum memory), the Majorana edge system density operator [16–28]. Yet solid-state realizations dynamics can monotonically relax to a plateau, or it undergoes of topological matter, such as topological superconductors acollapse-and-builduprelaxation[seeFig.2(c)]. Remarkably, [29–40], are often based on semiconductor nanostructures, when the memory depth increases above a critical value, the which are coupled to a structured phonon environment as edge correlation can nearly fully build up, corresponding to aresultofthedeformationofhostmaterialsanditslattice a new polaronic state of the topological superconductor in vibrations. In this case, the Markovian approximation and thus dynamical equilibrium with phonons. the Lindblad formalism usually fail. Compared to Markovian scenarios, key differences arise from the presence of quantum memory effects in non-Markovian processes: The information II. POLARONIC KITAEV CHAIN is lost from the system to the environment but flows back at a later time [41,42]. This generates a timescale into the Concretely, we consider the paradigmatic Kitaev p-wave system dynamics that is strictly absent in a Markovian context superconductor [43], with super-Ohmic coupling to a three- and raises the challenge as to what the unique dynamical dimensional (3D) structured phonon reservoir. The total consequences of this interplay between the topological state Hamiltonian is denoted by H0 Hk Hb (¯h 1). The N 1= +† ≡ of matter and non-Markovian dissipation are. Kitaev Hamiltonian Hk l −1 [( Jcl cl 1 !cl cl 1) = = − + + + + Here, we demonstrate that the quantum memory from H.c.] µ N c†c describes spinless fermions c and c† on a l 1 l l ! l l anon-Markovianparity-preservinginteractionofatopo- chain− of N sites= l,withanearest-neighbortunnelingamplitude logical p-wave superconductor with surrounding phonons J R,pairingamplitude! ! R,andchemicalpotentialµ. can give rise to intriguing edge mode relaxation dynam- When∈ µ < 2J and ! 0, it∈ is in the topological regime fea- | | ̸= ics without a Markovian counterpart. Our study is based turing unpaired Majorana edge modes γL/R j 1 fL/R, jb j = = on the polaron master equation, describing open-dissipative [here, we use the Majorana operators b c c† and 2 j 1 ! j j dynamics of a polaronic topological superconductor with † − = + b2 j i(c j c j )], with fL/R, j being exponentially localized near= the− left− (L)andright(R)edges.TheMajoranamodes exhibit a nonlocal correlation θ i γLγR 1thatcor- *[email protected] responds to the fermionic parity= of− the⟨ (degenerate)⟩=± ground 2469-9950/2020/102(24)/245115(8) 245115-1 ©2020 American Physical Society KAESTLE, SUN, HU, AND CARMELE PHYSICAL REVIEW B 102,245115(2020) Thus, the considered fermion-phonon interaction results in a polaronic Kitaev chain featuring phonon-dressed p-wave pair- ing, with phonon-induced quantum fluctuations. In deriving Eq. (1), we have ignored an arising energy renormalization 3 2 N † 2 term Hshift d k (gk/ωk )( l 1 cl cl ) ,knownasthepo- laron shift.= In the context of non-Markovian= dynamics, it FIG. 1. Left: A polaronic Kitaev chain, with phonon-dressed is standard practice" to neglect! this term if the initial condi- spinless fermions, exhibits a renormalized p-wave pairing B ! at tion is equilibrium [60,65], which is the case in the present temperature T [see Eq. (1)]. In the topological ground state,⟨ ⟩ two work, where the initial state is assumed to be the ground unpaired Majorana edge modes, γL and γR,emerge.Thecoupling state of an ideal Kitaev chain. However, to ensure validity gk to the structured phonon bath features mode dependence with of our results even in the presence of this energy renormal- spectral width σ . Right: Gaussian profile of gk in momentum space ization, in Appendix A we provide additional numerically for σ 0.2(green)andσ 0.6(red). = = exact calculations in the presence of the polaron energy shift, demonstrating that its inclusion does not affect the essential states. The entire chain is coupled to a 3D structured phonon physics presented below. Before tracing out the phononic degrees of freedom, we rewrite Eq. (1)asHp Hp,s Hp,I reservoir with a parity-preserving interaction, described by † = + + † N † † 3 3 Hp,b,withHp,b d k ωkrk rk for the reservoir. To recover Hb d k [ωkrk rk l 1 gkcl cl (rk rk )] [44–52]. Here, = = + = + the bare Kitaev Hamiltonian dynamics for the limiting case the operator r(†) annihilates (creates) phonons with momen- " k ! g 0, we introduce" a Franck-Condon renormalization of tum k and frequencies ω c k,wherec is the sound k k s s H →satisfying Tr [H ,ρ(t )] 0, with ρ(t )denotingthe velocity of the environment.= This coupling is commonly ap- p,I B p,I total density operator.{ The renormalized}= system Hamiltonian plied for the description of polaron formation [53–58]and H is given by in solid-state setups can represent the coupling of semicon- p,s ductor nanostructures to longitudinal acoustic phonons, which N 1 N − † † † † typically feature a long wavelength. We choose a generic Hp,s [ Jcl cl 1 ! B cl 1cl H.c.] µ cl cl , (2) = − + + ⟨ ⟩ + + − super-Ohmic fermion-phonon coupling gk which features fre- l 1 l 1 $= $= quency dependence modeled as a Gaussian function, gk where the pairing renormalization factor B is given explic- 2 2 2 = fph k/σ exp ( k /σ ), with σ being the width and fph be- ⟨ ⟩ − itly below. The system-reservoir interaction in the polaron pic- ing a dimensionless amplitude. We consider the topological N 1 2(R R† ) † † # ture reads Hp,I ! l −1 [(e− − B )cl 1cl H.c.]. superconductor and bath to be initially in equilibrium at low Wewillfocusonthelimit= = B 1,−⟨ which⟩ allows+ + us to treat temperatures, before a fast increase in the bath temperature ! ⟨ ⟩≪ Hp,I perturbatively in second-order Born theory, as dynamical induces an open-system dynamics. decoupling effects cannot occur [57,60]. In Appendix B,we provide a detailed derivation of the polaron master equation III. POLARON MASTER EQUATION for the reduced system density matrix ρS (t )ofthepolaron chain, obtaining At the heart of our following solution lies the polaron t representation of the coupled system (see Fig. 1). We de- 2 ∂t ρS (t ) i[Hp,s,ρS (t )] B dτ rive a master equation in second-order perturbation theory =− −⟨ ⟩ 0 of the dressed-state system-reservoir Hamiltonian [59]. In % ( cosh [φ(τ )] 1 [Xa, Xa( τ )ρS (t )] this dressed-state picture, the coherent process (i.e., higher- × { − } − sinh [φ(τ )][Xb, Xb( τ )ρS (t )] H.c.). (3) order contributions) from the fermion-phonon interaction is − − + retained through phonon-renormalized Hamiltonian parame- N 1 † † Here, Xa ! l −1 (cl cl 1 cl 1cl )andXb ters [53–58,60–64]. In this way, we efficiently account for the N 1 † †=− = + + + = ! − (c c c c )denotecollectivesystem non-Markovian character of the dynamics in the long-time l 1 l l 1 !l 1 l operators,= whose+ − dynamics+ obeys a time-reversed unitary limit not accessible and not captured in the typical second- evolution! governed by the renormalized Hamiltonian H , order Born approximation of the bare-interaction Hamiltonian p,s X ( τ ) e iHp,sτ X eiHp,sτ . φ(τ )representsthephonon H [44–47]. a,b − a,b b correlation− ≡ function, Defining collective bosonic operators R† d3k (g /ω )r†,weapplyapolarontransformation= k k k φ(τ ) d3k 2g (σ )/ω 2 N † † † (R R† ) † k k Up exp[ l 1 cl cl (R R)] resulting in cl e− − cl , = | | " = = − → % which describes phonon dressing of fermions. The h¯ωk transformed! total Hamiltonian H U H U 1 is derived coth cos (ωkτ ) i sin(ωkτ ) , (4) p p 0 p− × 2k T − as ≡ & ' B ( ) with kB being the Boltzmann constant.